Simulations of composite laminates inter and intra-laminar failure using on a non-local mean-field damage-enhanced multi-scale method

Computational & Multiscale

Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

Simulations of composite laminates inter- and intra-laminar

failure using on a non-local mean-field damage-enhanced

multi-scale method

Ling Wu (CM3), L. Adam (e-Xstream), B. Bidaine (e-Xstream), Ludovic Noels. (CM3) Experiments: F. Sket (IMDEA), J.M. Molina (IMDEA), A. Makradi (List)

STOMMMAC The research has been funded by the Walloon Region under the agreement no 1410246-STOMMMAC (CT-INT 2013-03-28) in the context of M-ERA.NET Joint Call 2014.

SIMUCOMP The research has been funded by the Walloon Region under the agreement no 1017232 (CT-EUC 2010-10-12) in the context of the ERA-NET +, Matera + framework.

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 2

•

Introduction

– Failure of composite laminates – Multi-scale modelling

– Mean-Field-Homogenization (MFH)

•

Micro-scale modelling

– Incremental-Secant MFH

– Damage-enhanced incremental-secant MFH

•

Multi-scale method for the failure analysis of composite

laminates

– Intra-laminar failure: Non-local damage-enhanced mean-field-homogenization

– Inter-laminar failure: Hybrid DG/cohesive zone model – Experimental validation

•

Introduction of uncertainties

– As a random field

Failure of composite laminates

•

Difficulties

– Different involved mechanisms at different scales

• Inter-laminar failure • Intra-laminar failure

– Direct finite element simulation On Micro-scale volume

Not possible at structural scale

Delamination Matrix rupture Pull out Bridging Fiber rupture Debonding

CM3

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Failure of composite laminates

•

Difficulties

– Different involved mechanisms at different scales

• Inter-laminar failure • Intra-laminar failure

– Direct finite element simulation is not possible at structural scale

– Continuum damage models do not represent accurately the intra-laminar failure

• Damage propagation direction is not in agreement with experiments

Delamination Matrix rupture Pull out Bridging Fiber rupture Debonding

Failure of composite laminates

•

Difficulties

– Different involved mechanisms at different scales

• Inter-laminar failure • Intra-laminar failure

– Direct finite element simulation is not possible at structural scale

– Continuum damage models do not represent accurately the intra-laminar failure

• Damage propagation direction is not in agreement with experiments

•

Solution:

– Embed damage model in a multi-scale formulation

– For computational efficiency: use of mean-field-homogenization

– For macro cracks: using hybrid DG/Cohesive zone model

Delamination Matrix rupture Pull out Bridging Fiber rupture Debonding

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•

Mean-Field-Homogenization

– Macro-scale • FE model

• At one integration point e is know, s is sought

– Transition

• Downscaling: e is used as input of the MFH model • Upscaling: s is the output of the MFH model

– Micro-scale

• Semi-analytical model

• Predict composite meso-scale response • From components material models

Multi-scale modelling

w

_I

w

₀

ε

σ

ε

σ





Mori and Tanaka 73, Hill 65, Ponte Castañeda 91, Suquet 95, Doghri et al 03, Lahellec et al. 11, Brassart et al. 12, …

•

Key principles

– Based on the averaging of the fields

– Meso-response

• From the volume ratios ( )

• One more equation required

– Difficulty: find the adequate relations

Mean-Field-Homogenization





V

a

V

V

a

1

(

X

)

d

1

I 0

 v



v

I I 0 0 I 0 I 0

σ

σ

σ

σ

σ

σ





v

_w



v

_w



v



v

I I 0 0 I 0 I 0

ε

ε

ε

ε

ε

ε





v

_w



v

_w



v



v

 

_I I

ε

σ



f

 

0 0

ε

σ



f

0 I

B

: ε

ε



e

?

e

B

0 I

B

: ε

ε



e

w

_I

w

₀ matrix inclusions composite _?

σ

ε

CM3

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•

Key principles (2)

– Linear materials

• Materials behaviours

• Mori-Tanaka assumption • Use Eshelby tensor

with

Mean-Field-Homogenization

w

_I

w

₀ matrix inclusions composite _?

σ

ε

1 0 1 1 0

:

(

)]

:

[











I

S

C

C

C

B

e



0 I



0 I

B

I

,

C

,

C

:

ε

ε



e 0

ε

ε 

 I I I

C

: ε

σ



0 0 0

C

: ε

σ



•

Key principles (2)

– Linear materials

• Materials behaviours

• Mori-Tanaka assumption • Use Eshelby tensor

with

– Non-linear materials

• Define a Linear Comparison Composite (LCC) • Common approach: incremental tangent

Mean-Field-Homogenization





0 alg alg I

B

I

,

C

₀

,

C

_I

:

ε

ε







e inclusions composite

σ

ε

alg I

C

alg 0

C

matrix:

σ

₀ I

ε





ε



ε

₀

w

_I

w

₀ matrix inclusions composite _?

σ

ε

1 0 1 1 0

:

(

)]

:

[











I

S

C

C

C

B

e



0 I



0 I

B

I

,

C

,

C

:

ε

ε



e 0

ε

ε 

 I I I

C

: ε

σ



0 0 0

C

: ε

σ



CM3

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Content

•

Micro-scale modelling

– Incremental-Secant Mean-Field-Homogenization (MFH) – Damage-enhanced incremental-secant MFH

Incremental-secant mean-field-homogenization

•

Material model

– Elasto-plastic material • Stress tensor

• Yield surface

• Plastic flow & • Linearization



,

p





eq





R

 

p



0

f

σ

σ

s

Y

)

(

:

pl el

_ε

_ε

C

σ





N

ε





p



pl

σ

N







f

σ

ε

el C pl

ε

ε

C

σ







alg

:

CM3

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•

New incremental-secant approach

– Perform a virtual elastic unloading from previous solution

• Composite material unloaded to reach the stress-free state

• Residual stress in components

New Linear Comparison Composite (LCC)

Incremental-secant mean-field-homogenization

inclusions composite

σ

ε

matrix unload I

ε



unload 0

ε



unload

ε



•

New incremental-secant approach

– Perform a virtual elastic unloading from previous solution

• Composite material unloaded to reach the stress-free state

• Residual stress in components

New Linear Comparison Composite (LCC)

– Apply MFH from unloaded state • New strain increments (>0)

• Use of secant operators

Incremental-secant mean-field-homogenization

inclusions composite

σ

ε

matrix unload I

ε



unload 0

ε



unload

ε







r 0 Sr Sr r I

B

I

,

C

₀

,

C

_I

:

ε

ε







e unload I/0 I/0 r I/0

ε

ε

ε











inclusions composite

σ

ε

matrix r I

ε





_ε

r r 0

ε



Sr I

C

Sr 0

C

CM3

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•

Zero-incremental-secant method

– Continuous fibres • 55 % volume fraction • Elastic – Elasto-plastic matrix

– For inclusions with high hardening (elastic) • Model is too stiff

Incremental-secant mean-field-homogenization

0 2 4 6 8 10 12 0 0.002 0.004

s/s

Y0

e

Longitudinal tension

FE (Jansson, 1992) C₀Sr 0 0.5 1 1.5 2 2.5 3 0 0.003 0.006 0.009

s/s

Y0 e

Transverse loading

FE (Jansson, 1992) C₀Sr inclusions composite

σ

ε

matrix r I

ε





_ε

r r 0

ε



Sr I

C

Sr 0

C

 

,

p



eq





R

 

p



0

f

σ

σ

s

Y

is underestimated

eq

σ

•

Zero-incremental-secant method (2)

– Continuous fibres

• 55 % volume fraction • Elastic

– Elasto-plastic matrix

– Secant model in the matrix

• Modified if negative residual stress

Incremental-secant mean-field-homogenization

0 2 4 6 8 10 12 0 0.002 0.004

s/s

Y0

e

Longitudinal tension

FE (Jansson, 1992) C₀Sr C₀S0 0 0.5 1 1.5 2 2.5 3 0 0.003 0.006 0.009

s/s

Y0 e

Transverse loading

FE (Jansson, 1992) C₀Sr C₀S0 inclusions composite

σ

ε

matrix r I

ε





_ε

r r 0

ε



Sr I

C

Sr 0

C

inclusions composite

σ

ε

matrix r I

ε





_ε

r r 0

ε



Sr I

C

S0 0

C

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 18

•

Verification of the method

– Spherical inclusions • 17 % volume fraction • Elastic – Elastic-perfectly-plastic matrix – Non-proportional loading

Incremental-secant mean-field-homogenization

.000 .020 .040 .060 .080 .100 0 10 20 30 40

e

t e₁₃e₂₃ e₃₃2e₁₁2e₂₂ -70 -45 -20 5 30 0 10 20 30 40

s

13 [Mpa ] t FE (Lahellec et al., 2013) MFH, incr. tg.

MFH, var. (Lahellec et al., 2013) MFH, incr. sec. -80 -40 0 40 80 120 0 10 20 30 40

s

33 [Mpa ] t FE (Lahellec et al., 2013) MFH, incr. tg.

MFH, var. (Lahellec et al., 2013) MFH, incr. sec.

FFT

Non-local damage-enhanced MFH

•

Material models

– Elasto-plastic material • Stress tensor

• Yield surface

• Plastic flow & • Linearization



,

p





eq





R

 

p



0

f

σ

σ

s

Y

)

(

:

pl el

_ε

_ε

C

σ





N

ε





p



pl

σ

N







f

σ

ε

el C pl

ε

ε

C

σ







alg

:

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 20

Non-local damage-enhanced MFH

•

Material models

– Elasto-plastic material • Stress tensor • Yield surface

• Plastic flow & • Linearization

– Local damage model

• Apparent-effective stress tensors

• Plastic flow in the effective stress space • Damage evolution



,

p





eq





R

 

p



0

f

σ

σ

s

Y

)

(

:

pl el

_ε

_ε

C

σ





N

ε





p



pl

σ

N







f

σ

ε

el C pl

ε

σ

ε

el

C

pl

ε

σˆ





σ

σ



1 D



ˆ





el

1



D

C

ε

C

σ







alg

:





σ

σ



1 D



ˆ

)

,

(

p

F

D



_D





ε

Non-local damage-enhanced MFH

•

Material models

– Elasto-plastic material • Stress tensor

• Yield surface

• Plastic flow & • Linearization

– Local damage model

• Apparent-effective stress tensors

• Plastic flow in the effective stress space • Damage evolution

– Non-Local damage model • Damage evolution

• Anisotropic governing equation

• Linearization



,

p





eq





R

 

p



0

f

σ

σ

s

Y

)

(

:

pl el

_ε

_ε

C

σ





N

ε





p



pl

σ

N







f

σ

ε

el C pl

ε

σ

ε

el

C

pl

ε

σˆ





σ

σ



1 D



ˆ





el

1



D

C

ε

C

σ







alg

:





σ

σ



1 D



ˆ

)

,

(

p

F

D



_D





ε

)

~

,

(

p

F

D



_D





ε



p



p

p











~



~

g

c





p

p

F

F

D

ˆ

D

:

ˆ

_~

D

~

1

alg

































ε

σ

ε

σ

C

σ

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 22

•

Equations summary: zero-approach

– For soft matrix response

• Remove residual stress in matrix • Avoid adding spurious internal energy

– Solve iteratively the system

– With the stress tensors

Non-local damage-enhanced MFH

unload I I r I

ε

ε

ε











unload 0 0 r 0

ε

ε

ε















r 0 Sr I 0 S 0 r I

B

I

,

(

1

)

C

,

C

:

ε

ε









e

D

matrix: inclusions composite

σ

ε

matrix:

ˆσ

₀ 0

σ

r I

ε





_ε

r r 0

ε



Sr I

C

S0 0

C

r 0 S0 0 0

(

1

)

C

:

ε

σ





D



r I Sr I res I I

σ

C

: ε

σ







I I 0 0

σ

σ

σ



v



v

) r ( I I ) r ( 0 0 ) r (

_ε

_ε

ε











v

v

•

Mesh-size effect

– Fictitious composite • 30%-UD fibres

• Elasto-plastic matrix with damage – Notched ply

Non-local damage-enhanced MFH

0 50 100 150 200 250 300 350 400 0 0.2 0.4 0.6 0.8 1 1.2 Fo rce [N/m m] Displacement [mm] Mesh size: 0.43 mm Mesh size: 0.3 mm Mesh size: 0.15 mm Mesh size: 0.1 mm

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 25

•

Multi-scale method for the failure analysis of composite

laminates

– Intra-laminar failure: Non-local damage-enhanced mean-field-homogenization

– Inter-laminar failure: Hybrid DG/cohesive zone model – Experimental validation

•

Weak formulation of a composite laminate

– Strong form

for each homogenized ply Ω_𝑖

for the matrix phase

– Boundary conditions

– Macro-scale finite-element discretization

Intra-laminar failure: Non-local damage-enhanced MFH

0

f

σ









T a a p

N

p

~

p

~

~



T

n

σ







p



p

p











~



~

g

c



_g





~





0



c

p

n

a a u

N u

u 















































p p p p u p p u u u

d

d

~ int ext ~ ~ ~ ~

~

_F

_F

F

F

p

u

K

K

K

K

 





 i i i i i i 1 L L   1     



i i    X X 1  2  3  5  4 



i i

w

w



CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 29

•

[45

o

4

/ -45

o4

]

S

- open hole laminate

– Tensile test on several coupons

– Propagation of the damaged zones in agreement with the fibre direction

Experimental validation

40 220 300 4.68±0.05 39.60±0.35 O13

-45

o

_-ply

45

o

_-ply

•

[45

o

4

/ -45

o4

]

S

- open hole laminate (2)

– Predicted delamination zones in agreement with experiments – Tensile stress within 15 %

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 32

•

[90

o

_{/ 45}

o

_{/ -45}

o

_{/ 90}

o

_{/ 0}

o

_]

S

- open hole laminate (2)

– Propagation of the damaged zones in agreement with the fibre direction

Experimental validation

•

[90

o

_{/ 45}

o

_{/ -45}

o

_{/ 90}

o

_{/ 0}

o

_]

S

- open hole laminate (3)

– Predicted delamination zones in agreement with experiments

Experimental validation

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 36

•

[45

o

_{/ -45}

o

_]

S

laminate under uniform tension

– No hole to trigger localization

– Material defects trigger localization

Introduction of uncertainties

Uniform volume fraction

5% variation in volume fraction

GFRP [45/-45]

•

Multi-scale method for the failure analysis of composite laminates

– Damage-enhanced MFH – Non-local implicit formulation

– Hybrid DG/CZM for delamination

•

Experimental validation

– Open-hole laminates

– Different stacking sequences

•

Introduction of material uncertainties in the model

– First simulations